Pole Lengths and Crossing Height.

This is related to some product design work I am doing and not being particular competent at maths have struggled to work out necessary steps in this problem. In summary, the problem is this. Two straight poles (P1 and P2) of length L1 and L2 are stuck in the ground a distance D apart (along the ground). P1 is at angle A1 to the vertical and P2 is at an angle A2 to the vertical. Two further poles (P3 and P4) are attached to P1 and P2. Pole P3 attaches at the bottom of P1 and attaches to the top of P2. Pole P4 attaches to the top of P1 and attaches to the bottom of P2. Poles P3 and P4 therefore cross each other at some point along their lengths. I simply wish to understand the basic steps in calculation I need to take to calculate the lengths of poles P3 and P4 and additionally the point at which they cross each other in terms of the point on pole P3 and the point on pole P4 that they cross. I thought this would be a straight forward problem to solve - it is easy to draw and therefore solve the problem diagrammatically - but I need to change values and therefore drawing is not feasible.

I would be grateful for any help. Basic steps in the calculations needed would be great. I can then plug these into a spreadsheet!

Re: Pole Lengths and Crossing Height.

Many Thanks for the interest. I have compiled the attached spreadsheet which includes 2 worksheets. The actual problem has different scenarios, but specifically the worksheet titled "query2" relates to the forum question.

Advice on "query1" would also be very gratefully received. My drawing is not the best. In "query1" drawing the right angled line running from top left to bottom right should just touch the end of the line P3.

Re: Pole Lengths and Crossing Height.

Scenario 1
There is a fair bit of calculation involved but the route through is reasonably straight forward.
The angle between P1 and D is known (90 - A1) so the cosine rule can be used to calculate P4. Similarly the angle between P2 and D is known so P3 can be calculated.
Knowing P3 and P4 it is now possible to calculate the angle between P3 and D, (using the sine rule in the triangle containing P2, P3 and D), and (similarly) the angle between P4 and D. Having these angles the height of the crossover between P3 and P4 can be calculated after which the lengths along P3 and P4 to the crossover point can be calculated.

Scenario 2
This is potentially more awkward because the first of the triangles to be used is a 'two sides and a none included angle' type. This gives rise to two possible solutions. In practice though it isn't a problem.
Start with the triangle made up of P2, P3 and D. The angle between P2 and D is known, (90 - A2) so the cosine rule can be used, in effect to calculate P3. What you have though is a quadratic equation in D which can be solved using the usual formula. You will take the solution with the positive sign in the middle.
Having D, using the cosine rule in the triangle made up of P1, D and P4 will give you P4.
You can then continue as in scenario 1.

Re: Pole Lengths and Crossing Height.

I have progressed some way now, but am struggling with the maths for the cross over points. Please can you confirm if the following is on the right lines and if so how best to simplify.

variable meaning
C1 vertical height of cross over points of poles above horizontal (ground) ie Line D
A3 angle of pole P3 with Line D
A4 angle of pole P4 with Line D
D horizontal distance between base of P1&P3 and P2&P4
d5 horizontal distance from base of P1&P3 to point vertically below cross over points of poles C1
A3 angle Pole P3 has with horizontal line D
A4 angle Pole P4 has with horizontal line D

Re: Pole Lengths and Crossing Height.

Actually, there is an alternative method.
Knowing the values of A3 and A4, you can calculate the angle between the P3 and P4 poles at the crossover point. Then, the sine rule can be used in the triangle made up of D, 'part of P3' and 'part of P4' to calculate the lengths up the P3 and P4 poles to the crossover point. A further (small) calculation would then be needed to calculate C1.
Do it both ways and see that you get the same results !